Contents eface x i troduction 1 apter 1 . Lyapuno v Stabilit y Theor y o f Differential Equation s 5 1.1. Lyapuno v Exponent s fo r Differentia l Equation s 6 1.2. Abstrac t Theor y o f Lyapunov Exponent s 9 1.3. Forwar d an d Backwar d Regularit y 1 6 1.4. Stabilit y Theor y o f Nonautonomous Differentia l Equation s 2 6 1.5. Lyapuno v Regularit y an d th e Oseledet s Decompositio n 3 1 apter 2 . Element s o f Nonuniform Hyperboli c Theor y 3 5 2.1. Dynamica l System s wit h Nonzer o Lyapuno v Exponent s 3 6 2.2. Nonunifor m Hyperbolicit y an d Regula r Set s 4 5 2.3. Holde r Continuit y o f Invariant Distribution s 4 8 2.4. Proo f o f the Multiplicativ e Ergodi c Theore m 5 1 apter 3 . Example s o f Nonuniformly Hyperboli c System s 6 1 3.1. Anoso v Diffeomorphism s 6 1 3.2. Diffeomorphism s wit h Nonzer o Lyapuno v Exponent s o n Surfaces 6 6 3.3. A Flow with Nonzer o Lyapuno v Exponent s 7 1 3.4. Geodesi c Flow s o n Compac t Manifold s o f Nonpositiv e Curvature 7 4 apter 4 . Loca l Manifol d Theor y 8 1 4.1. Existenc e o f Local Stabl e Manifold s 8 1 4.2. Basi c Propertie s o f Stabl e an d Unstabl e Manifold s 9 4 4.3. Absolut e Continuit y Propert y 9 9 4.4. Computin g th e Jacobia n o f the Holonom y Ma p 10 9 4.5. Partia l Hyperbolicit y 11 1 hapter 5 . Ergodi c Propertie s o f Smoot h Hyperboli c Measure s 11 5 5.1. Absolut e Continuit y an d Smoot h Invarian t Measure s 11 5 5.2. Ergodicit y o f Smo 5.1. Absolute Continuity and Smooth Invariant Measuresoth Hyperboli c Mea 5.3. Loca l Ergodicit y 12 2 IX
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